Solve for $x$ : $ 4|x - 4| + 2 = -1|x - 4| + 6 $
Add $ {1|x - 4|} $ to both sides: $ \begin{eqnarray} 4|x - 4| + 2 &=& -1|x - 4| + 6 \\ \\ { + 1|x - 4|} && { + 1|x - 4|} \\ \\ 5|x - 4| + 2 &=& 6 \end{eqnarray} $ Subtract ${2}$ from both sides: $ \begin{eqnarray} 5|x - 4| + 2 &=& 6 \\ \\ { - 2} &=& { - 2} \\ \\ 5|x - 4| &=& 4 \end{eqnarray} $ Divide both sides by ${5}$ $ \dfrac{5|x - 4|} {{5}} = \dfrac{4} {{5}} $ Simplify: $ |x - 4| = \dfrac{4}{5}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 4 = -\dfrac{4}{5} $ or $ x - 4 = \dfrac{4}{5} $ Solve for the solution where $x - 4$ is negative: $ x - 4 = -\dfrac{4}{5} $ Add ${4}$ to both sides: $ \begin{eqnarray} x - 4 &=& -\dfrac{4}{5} \\ \\ {+ 4} && {+ 4} \\ \\ x &=& -\dfrac{4}{5} + 4 \end{eqnarray} $ Change the ${ + 4}$ to an equivalent fraction with a denominator of $5$ $ x = - \dfrac{4}{5} {+ \dfrac{20}{5}} $ $ x = \dfrac{16}{5} $ Then calculate the solution where $x - 4$ is positive: $ x - 4 = \dfrac{4}{5} $ Add ${4}$ to both sides: $ \begin{eqnarray} x - 4 &=& \dfrac{4}{5} \\ \\ {+ 4} && {+ 4} \\ \\ x &=& \dfrac{4}{5} + 4 \end{eqnarray} $ Change the ${ + 4}$ to an equivalent fraction with a denominator of $5$ $ x = \dfrac{4}{5} {+ \dfrac{20}{5}} $ $ x = \dfrac{24}{5} $ Thus, the correct answer is $x = \dfrac{16}{5} $ or $x = \dfrac{24}{5} $.